Can anyone help explain why the following statistics are either Sufficient or not Sufficient?
Mode, Mean, Median, Standard Deviation, Skewness, Kurtosis, Range, IQR
A statistic is not 'sufficient' on its own. Sufficiency is with respect to a distribution (or rather a family).
For example, Wikipedia's article on sufficient statistics has
A statistic $T(X)$ is sufficient for underlying parameter $θ$ precisely if the conditional probability distribution of the data $X$, given the statistic $T(X)$, does not depend on the parameter $θ$
So a statistic may be sufficient for some parameter for one distribution but not sufficient for another parameter of another distribution. I can think of a number of distributions in which the sample mean is sufficient for some statistic ... and vastly more where it isn't.
You don't specify either parameters nor distributions in your question, so we can't really say whether any of them are sufficient. It depends on the situation.